Abstract
We construct double cross biproduct and bi-cycle bicrossproduct Lie bialgebras from braided Lie bialgebras. The main results generalize Majid's matched pair of Lie algebras and Drinfeld's quantum double and Masuoka's cross product Lie bialgebras.
Highlights
As an infinitesimal or semiclassical structures underlying the theory of quantum groups, the notion of Lie bialgebras was introduced by Drinfeld in his remarkable report [3], where he introduced the double Lie bialgebra D(g) as an important construction
There is a close relation between extension theory and cross product Lie bialgebras, see Masuoka [7]
The concept of Yetter-Drinfeld modules over Lie bialgebras was introduced by Majid in [6], which he used to construct biproduct Lie bialgebras
Summary
As an infinitesimal or semiclassical structures underlying the theory of quantum groups, the notion of Lie bialgebras was introduced by Drinfeld in his remarkable report [3], where he introduced the double Lie bialgebra D(g) as an important construction. There is a close relation between extension theory and cross product Lie bialgebras, see Masuoka [7]. Majid’s and Masuoka’s results will be generalized as corollaries of our main results. Throughout this paper, all vector spaces will be over a fixed field of character zero. The identity map of a vector space V is denoted by idV : V → V
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